Topic 4: Flow Analysis

1
Topic 4 Flow Analysis
Some slides come from Prof. J. N. Amaral
(amaral_at_cs.ualberta.ca)
2
Topic 4 Flow Analysis

• Motivation
• Control flow analysis
• Dataflow analysis
• Advanced topics

3
Reading List

• Slides
• Dragon book chapter 8.4, 8.5, Chapter 9
• Muchnicks book Chapter 7
• Other readings as assigned in class or homework

4
Flow Analysis
Control flow analysis
Interprocedural
Program
Intraprocedural
Procedure
Flow analysis
Data flow analysis
Local
Basic block

• Control Flow Analysis - Determine control
  structure of a program and build a Control Flow
  Graph.
• Data Flow analysis - Determine the flow of scalar
  values and ceretain associated properties
• Solution to the Flow analysis Problem
  propagation of data flow information along a flow
  graph.

5
Introduction to Code Optimizations

• Code optimization - a program transformation
  that preserves correctness and improves the
  performance (e.g., execution time, space, power)
  of the input program. Code optimization may be
  performed at multiple levels of program
  representation
• 1. Source code
• 2. Intermediate code
• 3. Target machine code
• Optimized vs. optimal - the term optimized is
  used to indicate a relative performance
  improvement.

6
Motivation

• S1 A 2 (def of A)
• S2 B 10 (def of B)
• S3 C A B determine if C is a constant 12?
• S4 Do I 1, C
• AI BI DI-1

. . .
7
Basic Blocks
A basic block is a sequence of consecutive
intermediate language statements in which flow
of control can only enter at the beginning and
leave at the end.

• Only the last statement of a basic block can be
  a branch statement and only the first statement
  of a basic block can be a target of a branch.
  However, procedure calls may need be treated with
  care within a basic block (Procedure call starts
  a new basic block)

(AhoSethiUllman, pp. 529)
8
Basic Block Partitioning Algorithm

• 1. Identify leader statements (i.e. the first
  statements of basic blocks) by using the
  following rules
• (i) The first statement in the program is a
  leader
• (ii) Any statement that is the target of a
  branch statement is a leader (for most ILs.
  these are label statements)
• (iii) Any statement that immediately follows a
  branch or return statement is a leader
• 2. The basic block corresponding to a leader
  consists of the leader, and all statements up to
  but not including the next leader or up to the
  end of the program.

9
Example
The following code computes the inner product of
two vectors.

• (1) prod 0
• (2) i 1
• (3) t1 4 i
• (4) t2 at1
• (5) t3 4 i
• (6) t4 bt3
• (7) t5 t2 t4
• (8) t6 prod t5
• (9) prod t6
• (10) t7 i 1
• (11) i t7
• if i lt 20 goto (3)
• (13)

begin prod 0 i 1 do begin
prod prod ai bi i i 1
end while i lt 20 end
Source code.
Three-address code.
10
Example
The following code computes the inner product of
two vectors.
(1) prod 0 (2) i 1 (3) t1 4
i (4) t2 at1 (5) t3 4 i (6) t4
bt3 (7) t5 t2 t4 (8) t6 prod
t5 (9) prod t6 (10) t7 i 1 (11) i
t7 (12) if i lt 20 goto (3) (13)
Rule (i)
begin prod 0 i 1 do begin
prod prod ai bi i i 1
end while i lt 20 end
Source code.
Three-address code.
11
Example
The following code computes the inner product of
two vectors.
Rule (i)
(1) prod 0 (2) i 1 (3) t1 4
i (4) t2 at1 (5) t3 4 i (6) t4
bt3 (7) t5 t2 t4 (8) t6 prod
t5 (9) prod t6 (10) t7 i 1 (11) i
t7 (12) if i lt 20 goto (3) (13)
begin prod 0 i 1 do begin
prod prod ai bi i i 1
end while i lt 20 end
Rule (ii)
Source code.
Three-address code.
12
Example
The following code computes the inner product of
two vectors.
(1) prod 0 (2) i 1 (3) t1 4
i (4) t2 at1 (5) t3 4 i (6) t4
bt3 (7) t5 t2 t4 (8) t6 prod
t5 (9) prod t6 (10) t7 i 1 (11) i
t7 (12) if i lt 20 goto (3) (13)
Rule (i)
begin prod 0 i 1 do begin
prod prod ai bi i i 1
end while i lt 20 end
Rule (ii)
Rule (iii)
Source code.
Three-address code.
13
Example
B1
(1) prod 0 (2) i 1
(3) t1 4 i (4) t2 at1 (5) t3 4
i (6) t4 bt3 (7) t5 t2 t4 (8)
t6 prod t5 (9) prod t6 (10) t7 i
1 (11) i t7 (12) if i lt 20 goto (3)
B2
Basic Blocks
B3
(13)
14
Transformations on Basic Blocks

• Structure-Preserving Transformations
• common subexpression elimination
• dead code elimination
• renaming of temporary variables
• interchange of two independent adjacent
  statements
• Others

15
Transformations on Basic Blocks

• The DAG representation of a basic block lets
  compiler perform the code-improving
  transformations on the codes represented by the
  block.

16
Transformations on Basic Blocks

• Algorithm of the DAG construction for a basic
  block
• Create a node for each of the initial values
  of the variables in the basic block
• Create a node for each statement s, label
  the node by the operator in the statement s, and
  attach the list of variables for which it is the
  last definition in the basic block.
• The children of a node N are those nodes
  corresponding to statements that are last
  definitions of the operands used in the statement
  associated with node N.

Tiger book pp533
17
An Example of Constructing the DAG

• Step
  (1) create node 4 and i0
• Step (2) create node
• Step
  (3) attach identifier t1
• Step (1) create
  nodes labeled , a
• Step (2) find previously node(t1)
• Step (3) attach label
• Here we determine that
• node (4) was created
• node (i) was created
• node () was created

t1 4i t2 at1 t3 4i
t1

i0
4
t2

t1,t3
a0

i0
4
just attach t3 to node
18
Example of Common Subexpression Elimination
c

a b c b a d c b c d b

1. a b c
2. b a d
3. c b c
4. d a - d

b,d
-
d0
a

c0
b0
If a node N represents a common subexpression, N
has more than one attached variables in the DAG.

• Detection
• Common subexpressions can be detected by
  noticing, as a new node m is about to be added,
  whether there is an existing node n with the same
  children, in the same order, and with the same
  operator.
• if so, n computes the same value as m and may be
  used in its place.

19
Example of Dead Code Elimination
if x is never referenced after the statement x
yz, the statement can be safely eliminated.
20
Example of Renaming Temporary Variables
rename
(1) t b c (1) u b c
t
u
Change (rename) label



b0
c0
b0
c0
if there is an statement t b c, we can
change it to u b c and change all uses of t
to u.
a code in which each temporary is defined only
once is called a single assignment form.
21
Example of Interchange of Statements
t1
t2

• t1 b c
• t2 x y

b0
c0
x0
y0

• Observation
• We can interchange the statements without
  affecting the value of the block if and only if
  neither x nor y is t1 and neither b nor c is t2,
  i.e. we have two DAG subtrees.

22
Example of Algebraic Transformations

• Arithmetic Identities
• x 0 0 x x
• x 0 x
• x 1 1 x x
• x / 1 x
• - Replace left-hand side with simples right
  hand side.
• Associative/Commutative laws
• x (y z) (x y) z
• x y y x

• Reduction in strength
• x 2 x x
• 2.0 x x x
• x / 2 x 0.5
• - Replace an expensive operator with a cheaper
  one.
• Constant folding
• 2 3.14 6.28
• -Evaluate constant expression at compile time

23
Control Flow Graph (CFG)

• A control flow graph (CFG), or simply a flow
  graph, is a directed multigraph in which the
  nodes are basic blocks and edges represent flow
  of control (branches or fall-through execution).
• The basic block whose leader is the first
  statement is called the initial node or start
  node
• There is a directed edge from basic block B1
  to basic B2 in the CFG if
• (1) There is a branch from the last statement of
  B1 to the first
• statement of B2, or
• (2) Control flow can fall through from B1 to B2
  because B2
• immediately follows B1, and B1 does not
  end with an
• unconditional branch

24
Example
(1) prod 0 (2) i 1
B1
Control Flow Graph
Rule (2)
(3) t1 4 i (4) t2 at1 (5) t3 4
i (6) t4 bt3 (7) t5 t2 t4 (8)
t6 prod t5 (9) prod t6 (10) t7 i
1 (11) i t7 (12) if i lt 20 goto (3)
B2
B1
B2
B3
Rule (1)
Rule (2)
(13)
B3
25
CFGs are Multigraphs
Note there may be multiple edges from one basic
block to another in a CFG. Therefore, in
general the CFG is a multigraph. The edges are
distinguished by their condition labels. A
trivial example is given below
101 . . . 102 if i gt n goto L1
Basic Block B1
False
True
103 label L1 104 . . .
Basic Block B2
26
Identifying loops
Question Given the control flow graph of a
procedure, how can we identify loops?
Answer We use the concept of dominance.
27
Dominators

• Node (basic block) D in a CFG dominates node N
  if every path from the start node to N goes
  through D. We say that node D is a dominator of
  node N.
• Define DOM(N) set of node Ns dominators, or
  the dominator set for node N.
• Note by definition, each node dominates itself
  i.e., N ? DOM(N).

28
Domination Relation

• Definition Let G (N, E, s) denote a
  flowgraph.
• and let n, n ? N.
• 1. n dominates n, written n ? n
• each path from s to n contains n.
• 2. n properly dominates n, written n lt n
• n ? n and n ? n.
• 3. n directly (immediately) dominates n, written
  n ltd n
• n lt n and
• there is no m ? N such that n lt m lt
  n.
• 4. DOM(n) n n ? n is the set of
  dominators of n.

29
Domination Property

• The domination relation is a partial ordering
• Reflexive
• A ? A
• Antisymmetric
• A ? B B ? A
• Transitive
• A ? B and B ? C A ? C

30
Computing Dominators
Observe if a dominates b, then a b, or a
is the only immediate predecessor of b, or b
has more than one immediate predecessor, all of
which are dominated by a.
DOM(b) b U n DOM(p)
p ? pred(b)
Quiz why here is the intersection operator
instead of the union?
31
An Example

• Domination relation

(1, 1), (1, 2), (1, 3), (1, 4) (2, 3),
(2, 4), (2, 10)
S
1
2
3
Direct domination
4
1 ltd 2, 2 ltd 3,
5
6
7
DOM
8
DOM(1) 1 DOM(2) 1, 2 DOM(10)
1, 2, 10
9
10
DOM(8) ?
DOM(8) 1,2,3,4,5,8
32
Question

• Assume node m is an immediate dominator of a
  node n, is m necessarily an immediate predecessor
  of n in the flow graph?

S
1
2
3
4
5
6
7
Answer NO!
Example consider nodes 5 and 8.
8
9
10
33
Dominance Intuition
S
Imagine a source of light at the start node, and
that the edges are optical fibers
1
2
3
4
5
To find which nodes are dominated by a given
node a, place an opaque barrier at a and
observe which nodes became dark.
6
7
8
9
10
34
Dominance Intuition
S
The start node dominates all nodes in the
flowgraph.
1
2
3
4
5
6
7
8
9
10
35
Dominance Intuition
S
1
Which nodes are dominated by node 3?
2
3
4
5
6
7
8
9
10
36
Dominance Intuition
S
1
Which nodes are dominated by node 3?
2
3
4
Node 3 dominates nodes 3, 4, 5, 6, 7, 8, and 9.
5
6
7
8
9
10
37
Dominance Intuition
S
1
Which nodes are dominated by node 7?
2
3
4
5
6
7
8
9
10
38
Dominance Intuition
S
1
Which nodes are dominated by node 7?
2
3
4
5
Node 7 only dominates itself.
6
7
8
9
10
39
Immediate Dominators and Dominator Tree

• Node M is the immediate dominator of node N gt
  Node M must be the last dominator of N on any
  path from the start node to N.
• Therefore, every node other than the start node
  must have a unique immediate dominator (the start
  node has no immediate dominator.)

What does this mean ?
40
A Dominator Tree
A dominator tree is a useful way to represent the
dominance relation. In a dominator tree the
start node s is the root, and each node d
dominates only its descendants in the tree.
41
Dominator Tree (Example)
A flowgraph (left) and its dominator tree (right)
42
Natural Loops

• Back-edges - an edge (B, A), such that
• A lt B (A properly dominates B).
• Header --A single-entry node which dominates
  all nodes in a subgraph.
• Natural loops given a back edge (B, A), a
  natural loop of (B, A) with entry node A is the
  graph A plus all nodes which is dominated by A
  and can reach B without going through A.

43
Find Natural Loops
One way to find natural loops is
start
1) find a back edge (b,a)
a
2) find the nodes that are dominated by a.
3) look for nodes that can reach b among the
nodes dominated by a.
b
44
Algorithm to finding Natural Loops
Input A flow graph G and a back edge n -gt
d Output the natural loop of n -gtd Initial a
loop L with nodes n and d Ln, d Mark d as
visible so that the following search does not
reach beyond d Perform a depth-first search on
the revise control-flow graph starting with node
n Insert all the nodes visited in this search
into loop L.

• - Alg. 9.46 (Aho et. al., pp665)

45
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
6
5
7
8
10
9
46
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
(9,1)
Entire graph
4
6
5
7
8
10
9
47
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
(9,1)
Entire graph
4
6
5
7
8
10
9
48
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
(9,1)
Entire graph
4
6
5
7
8
10
9
49
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
(9,1)
Entire graph
4
(10,7)
7,8,10
6
5
7
8
10
9
50
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
(9,1)
Entire graph
4
(10,7)
7,8,10
6
5
7
8
10
9
51
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
(9,1)
Entire graph
6
5
(10,7)
7,8,10
7
(7,4)
8
10
9
52
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
6
5
(9,1)
Entire graph
(10,7)
7,8,10
7
(7,4)
4,5,6,7,8,10
8
10
9
53
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
(9,1)
Entire graph
6
5
(10,7)
7,8,10
7
(7,4)
4,5,6,7,8,10
(8,3)
8
10
9
54
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
(9,1)
Entire graph
6
5
(10,7)
7,8,10
7
(7,4)
4,5,6,7,8,10
(8,3)
8
10
9
55
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
(9,1)
Entire graph
6
5
(10,7)
7,8,10
7
(7,4)
4,5,6,7,8,10
(8,3)
3,4,5,6,7,8,10
8
10
9
56
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
(9,1)
Entire graph
6
5
(10,7)
7,8,10
7
(7,4)
4,5,6,7,8,10
(8,3)
3,4,5,6,7,8,10
8
(4,3)
10
9
57
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
(9,1)
Entire graph
6
5
(10,7)
7,8,10
7
(7,4)
4,5,6,7,8,10
(8,3)
3,4,5,6,7,8,10
8
(4,3)
10
9
58
An Example
Find all back edges in this graph and the natural
loop associated with each back edge
1
2
3
Back edge
Natural loop
4
(9,1)
Entire graph
6
5
(10,7)
7,8,10
7
(7,4)
4,5,6,7,8,10
(8,3)
3,4,5,6,7,8,10
8
(4,3)
3,4,5,6,7,8,10
10
9
59
Reducible Flow Graphs

• Def a CFG V, E is reducible iff E can be
  partitioned into two classes
• 1. Forward edges form an acyclic graph where
  every node is reachable from the initial node
• 2. Back edges
• Motivation Structured programs always
  reducible
• (note programs with gotos are still often
  reducible)
• Intuition No jumps into the middle of loops.

60
How to check if a graph G is reducible?

• Step1 compute dom relation
• Step2 identify all back edges
• Step3 remove all back edges and derive G
• Step4 check if G is acyclic
• Example

Bad cycle can be entered from 2 different
places
1
2
3
61
Loops in Reducible Flow Graphs

• Intuitive no bad loops.
• In fact, all loops in structured programs are
  natural loops
• In practice
• Structured programs only produce reducible graphs.

62
More About Loops

• Read the following slides and think?

63
Region and Loop
A region is a set of nodes N that include a
header with the following properties (i) the
header must dominate all the nodes in the
region (ii) All the edges between nodes in N are
in the region (except for some edges that enter
the header)
A loop is a special region that has the following
additional properties (i) it is strongly
connected (ii) All back edges to the header are
included in the loop
64
Loops in Control Flow Graphs

• Motivation Programs spend more time in loops, so
  there is a larger payoff from optimizations that
  exploit loop structure e.g., loop-invariant code
  motion, software pipelining, etc.
• Basic idea Identify regions (subgraphs) of the
  CFG that correspond to program loops. Use a
  general approach based on analyzing
  graph-theoretical properties of the CFG - uniform
  treatment for program loops written using
  different loop structures (e.g. while, for) and
  loops constructed out of gotos.

65
Relation of Flowgraph and Loop A Definition
A strongly-connected component (SCC) of a
flowgraph G (N, E, s) is a subgraph G
(N, E, s) in which there is a path from
each node in N to every node in N.

• A strongly-connected component G (N, E, s)
  of a flowgraph G (N, E, s) is a loop with entry
  s if s dominates all nodes in N.

66
A Strongly-Connected Graph But Not a Loop

• Counter-example not every strongly connected
  graph is a loop -- see the subgraph consists of
  nodes 2 ad 3

1
2
3
Why?
No node in the subgraph dominates all the other
nodes.